MATRICES AND ITS APPLICATIONS
ABSTRACT
This Project examines matrices and three of its applications. Matrix theories were used to solve economic problems, which involves methods at which goods can be produced efficiently. To encode and also to decode very sensitive information. This project work also goes further to apply matrices to solve a 3 x 3 linear system of equations using row reduction methods.
TABLE OF CONTENT
CHAPTER ONE: GENERAL INTRODUCTION
1.0 BACKGROUND OF THE STUDY
1.2 STATEMENT OF PROBLEM
1.3 AIMS AND OBJECTIVES
CHAPTER TWO
2.0 LITERATURE REVIEW
CHAPTER THREE: THEORY OF MATRICES
3.0 THEORY OF MATRICES
3.1.1 DEFINITION AND TYPES OF MATRICES
3.2 ADDITION AND SUBTRACTION OF MATRICES
3.3 PROPERTIES OF MATRIX ADDITION
3.4 SCALAR MULTIPLICATION
3.5 MULTIPLICATION OF MATRICES
3.6 PROPERTIES OF MATRIX MULTIPLICATION
3.7 ELEMENTARY ROW OPERATION
3.8 ECHELON AND ROW-REDUCED ECHELON FORMS OF MATRIX
3.9 DETERMINANT OF MATRIX
3.10 PROPERTIES OF DETERMINANT
3.11 INVERSE OF MATRIX
3.12 PROPERTIES OF INVERSE MATRIX
3.13 A METHOD OF COMPUTING THE INVERSE OF A MATRIX
CHAPTER FOUR: APPLICATIONS OF MATRICES
4.0 INTRODUCTION
4.1 APPLICATION OF MATRICES TO CRYPTOGRAPHY
4.2 APPLICATION TO ECONOMICS
4.2.1 OPEN AND CLOSE ECONOMIC SYSTEM
4.3 APPLICATION OF MATRIX TO SYSTEM OF LINEAR EQUATION
4.4 SOLVING A LINEAR SYSTEM USING (ROW REDUCTION) METHOD
CHAPTER FIVE: SUMMARY ,CONCLUSIONS
5.1 SUMMARY
5.2 CONCLUSIONS
REFERENC
CHAPTER ONE
GENERAL INTRODUCTION
1.0 BACKGROUND OF THE STUDY
In orderto unfold thehistory of Matrices and Its Applications,theinfluence of matrices inthe mathematicalworldis spreadwide becauseit providesan importantbaseto many of the principles and practices.It is importantthatwe firstdeterminewhatmatricesis. As such, this definitionis notacomplete andcomprehensiveanswer,but rather a broaddefinition looselywrapping itselfaround thesubject.
“Matrix” is the Latin word for womb, and it retains that sense in English. It can also mean more generally any place in which something is formed or produced.
The origin of mathematical matrices lies with the study of systems of simultaneous linear equations. An important Chinese text from between 300Bc and Ad 200, nine chapters of the mathematical art, gives the first known example of the use of matrix methods to solve simultaneous equations.(Laura Smoller (2012))[9]
In the treatises seventh chapter “too much and not enough”, the concept of a determinant first appears, nearly two milknna before its supposed inventions by the japanese mathematician SEKI KOWA in 1683 or his german contemporary GOTTFRIED LEIBNIZ (who is also credited wiith the invention of differential calculus, seperately from but simutaneously with isaac Newton).
More uses of matrix-like arrangements of numbers appears in which a method is given for solving simultaneous equations using a counting board that is mathematically identical to the modern matrix method of solution outlined by Carh Fridrich Gauss (1777-1855) also known as Gaussan Elimination.(Vitull marie 2012 )[17]
This projectseeks to give anoverview of thehistory of matricesand its practicalapplications touchingon the varioustopics used in concordancewith it.
Around 4000 years ago, thepeopleof Babylon knew how to solvea simple2X2 system of linear equations withtwo unknowns.Around 200 BC, theChinese publishedthat“Nine Chapters of the MathematicalArt,”they displayed theability tosolvea 3X3 systemof equations.(perotti)[13].Thepower andprogress inMatricesand its application did notcome tofruitionuntilthe late 17th century.
Theemergence of thesubjectcamefromdeterminants, values connectedto a square matrix,studied by thefounder of calculus, Leibnitz,in thelate17thcentury. Lagrangecameout withhis work regarding Lagrange multipliers, a way to“characterize themaxima andminima multivariate functions.” (Darkwing) Morethan fiftyyears later, Cramer presented his ideasof solvingsystemsof linearequationsbasedon determinants more than50 years afterLeibnitz (Darkwing). Interestinglyenough, Cramerprovided no proof for solvingan n x n system.
As mentioned before,Gauss work dealtmuch withsolvinglinear equationsthemselves initially,butdid nothave as muchto do with matrices. In order for matrix algebra todevelop,a propernotationor methodof describingthe process was necessary.Also vital tothis process was adefinition of matrix multiplicationand the facets involving it.“Theintroduction of matrix notationand the inventionof the word matrix were motivated by attemptsto developthe right algebraiclanguage for studying determinants. In 1848, J.J. Sylvester introduced theterm “matrix,” theLatinword for womb,as anamefor an array of numbers. He used womb, becausesee, linearalgebrahas becomemore relevantsince theemergence of calculus eventhough it’s foundationalequationof ax+ b=0 dates backcenturies.
Euler broughtto lightthe idea thata systemof equationsdoesn’tnecessarilyhave to have asolution. He recognized the needfor conditions tobe placed upon unknown variables inorder to finda solution.Theinitialwork up until this period mainlydealtwith theconceptof uniquesolutions and squarematrices where thenumber of equations matchedthe numberof unknowns.
With theturn intothe19thcenturyGauss introduced a procedure to beusedfor solvinga systemof linearequations. His work dealt mainly withthe linear equations and hadyet tobring inthe idea of matrices or their notations. His efforts dealtwithequations of differing numbers andvariablesas wellas thetraditionalpre-19thcentury works of Euler,Leibnitz, andCramer. Gauss’work is now summedup in the termGaussian elimination.This methoduses the concepts of combining, swapping, or multiplying rows with eachother inorder to eliminate variables fromcertain equations.Aftervariables are determined,thestudent is thento use back substitution tohelp findthe remainingunknown variables.
Heviewed a matrixas ageneratorof determinants(Tucker, 1993).Theother part,matrix multiplication or matrixalgebracamefrom thework ofArthurCayleyin 1855.
Cayley’s defined matrix multiplication as, “the matrix of coefficients for thecomposite transformationT2T1is theproductof the matrix forT2times thematrix ofT1”(Tucker, 1993). His work dealingwith Matrixmultiplicationculminatedin his theorem, the Cayley-Hamilton Theorem. Simply stated, a square matrixsatisfiesMatrices atthe end of the 19thcentury were heavily connectedwith Physics issues andfor mathematicians, moreattentionwas givento vectorsas they proved tobe basic mathematical elements.Withtheadvancementof technologyusing themethods of Cayley, Gauss, Leibnitz, Euler, andothers determinantsand linear algebra movedforwardmore quickly and more effective. Regardless of thetechnologythough Gaussian eliminationstillproves to be thebest way known to solve asystemof linear equations (Tucker,1993).
Theinfluence of matrices and it’s applicationsinthe mathematicalworldis spreadwide becauseit providesan importantbaseto many of the principles and practices. Some of thethings Matrices is usedfor are tosolvesystems of linearformat, to find least-square bestfitlines topredictfuture outcomes or find trends, to encode and decode messages. Other morebroad topicsthatit is used for areto solve questionsof energy in Quantummechanics. It is also usedto create simple everyday household games likeSudoku. Itis because of these practicalapplications thatMatriceshas spread so far and advanced.Thekey, however,is to understand that thehistory of linear algebra providesthe basis for theseapplications.
Althoughlinearalgebrais a fairly new subjectwhencomparedto othermathematical practices,its uses are widespread.Withthe efforts of calculussavvy Leibnitz theconceptof using systems of linear equations tosolveunknowns was formalized. Otherefforts fromScholars like Cayley.Euler, Sylvester, and others changed matricesinto theuseof linear algebratorepresent them. Gauss broughthis theory tosolve systems of equations proving to bethe most effectivebasis for solving unknowns.
Technologycontinues to push the use furtherand further, butthe historyof matrices and its applicationcontinues toprovide thefoundation. Even thoughevery few years companiesupdate their textbooks, the fundamentals stay thesame.(laura smoller (2001)[9].
1.2 STATEMENT OF PROBLEM
Due to the great need of security for passing sensitive information from one person to another or from one organization to another through electronic technology, there is need for cryptography as a solution to this problem.
Also in economics this research work is going to discuss how Leontief model is used to represent the economy as a system of linear equation so as to calculate the gross domestic products and goods production efficiently.
1.3 AIMS AND OBJECTIVES
i. To apply matrices to Cryptography, Economic Models and system of Linear Equations
ii.To improve the methods at which increase in production out-put can be achieved
iii.To show ways at which sensitive information can be passed across mathematically.
iv. To disseminate this improved methods to the relevant communities and end use
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